Finite Group Databasebeta

← #126987

Subgroups — #126987

Order 512.

1082 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1082 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1466C211yesmaximal, characteristic
2C2×#1465C211yesmaximal, characteristic
3C4×#386C211yesmaximal, characteristic
4#7187C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C23×C42C411yescharacteristic
7C2×#432—21no—
8C2×#407—21no—
9C2×#386C412yes—
10C2×C4×C22⋊C4—21no—
11C2×C4×C22⋊C4C411yescharacteristic
12#1466—22no—
13#1466C414yescomplemented
14#1465—22no—
15#1465C414yescomplemented
16C2×#324—21no—
17C2×#323—21no—
18C2×#323C2212yes—
19C2×#323C2212yes—
20#1437—21no—
21#1436—21no—
22C22⋊#95C2212yes—
23C4×#95—21no—
24C23×D4C2×C411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C412yes—
27C24×C4D411yescharacteristic
28C22×M16—41no—
29C2×C4×D4—41no—
30C2×C4×D4—41no—
31C2×C4×D4—42no—
32C2×C4×D4C2×C412yes—
33C2×C4×D4C2×C412yes—
34C2×C4×D4D412yes—
35C2×C4×D4Q812yes—
36C22×C4⋊C4D411yescharacteristic
37C22×C22⋊C4—22no—
38C22×C22⋊C4C2×C412yes—
39C22×C22⋊C4D411yescharacteristic
40C22×C22⋊C4C2311yescharacteristic
41C22×C42—21no—
42C22×C42—21no—
43C22×C42—21no—
44C22×C42—44no—
45C22×C42—41no—
46C22×C42—41no—
47C22×C42D411yescharacteristic
48#432—24no—
49#432—42no—
50C2×C4⋊C8—41no—
51#407—24no—
52#407—42no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—41no—
56C2×#97—41no—
57#386—24no—
58#386—24no—
59C4×C22⋊C4—44no—
60C4×C22⋊C4—42no—
61C2×#95—22no—
62C2×#95—21no—
63C2×#95—22no—
64C2×#95—22no—
65#324—24no—
66#323—24no—
67#323—22no—
68#323—22no—
69#323C2×C414yes—
70#323C2×C414yes—
71C25C22⋊C411yescharacteristic
72C25C22×C411yescharacteristic
73C22×D4—22no—
74C22×D4—22no—
75C22×D4—41no—
76C22×D4—41no—
77C22×D4—42no—
78C22×D4C4212yes—
79C22×D4C4⋊C412yes—
80C23×C4—22no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—42no—
90C23×C4—44no—
91C23×C4—42no—
92C23×C4—44no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4C4212yes—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C4⋊C412yes—
103C23×C4C4⋊C412yes—
104C23×C4M1614yescomplemented
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C2×M16—44no—
108C2×M16—82no—
109C2×M16—82no—
110C22×C8—42no—
111C22×C8—42no—
112C4×D4—42no—
113C4×D4—42no—
114C4×D4—42no—
115C4×D4—42no—
116C4×D4—42no—
117C4×D4—42no—
118C4×D4—42no—
119C4×D4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4C22⋊C412yes—
123C2×C4⋊C4C4⋊C412yes—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—44no—
131C2×C22⋊C4—44no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4C22⋊C412yes—
135C2×C22⋊C4C4⋊C412yes—
136C2×C22⋊C4M1614yescomplemented
137C2×C22⋊C4C22×C411yescharacteristic
138C2×C22⋊C4C22×C411yescharacteristic
139C2×C22⋊C4C2×D411yescharacteristic
140C2×C22⋊C4C2×Q811yescharacteristic
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—22no—
144C2×C42—24no—
145C2×C42—42no—
146C2×C42—44no—
147C2×C42—42no—
148C2×C42—48no—
149C2×C42—44no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42—44no—
153C2×C42—42no—
154C2×C42—42no—
155C2×C42—42no—
156C2×C42C22⋊C412yes—
157C2×C42C4⋊C412yes—
158C4⋊C8—44no—
159C22⋊C8—24no—
160C22⋊C8—24no—
161C22⋊C8—42no—
162C22⋊C8—42no—
163C22⋊C8—44no—
164#97—44no—
165#95—28no—
166#95—44no—
167#95—44no—
168#95—42no—
169C24—21no—
170C24—21no—
171C24—22no—
172C24—22no—
173C24—21no—
174C24—21no—
175C24—22no—
176C24—41no—
177C24—41no—
178C24—42no—
179C24—41no—
180C24—42no—
181C24—41no—
182C24#9512yes—
183C24C2×C4211yescharacteristic
184C24C2×C22⋊C411yescharacteristic
185C24C2×C4⋊C411yescharacteristic
186C24C2×M1612yes—
187C24C2×M1612yes—
188C2×D4—24no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—44no—
198C2×D4—44no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—48no—
226C22×C4—48no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—48no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—416no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—84no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4#9512yes—
262C22×C4#9512yes—
263C22×C4#9512yes—
264C22×C4#9912yes—
265C22×C4#9911yescharacteristic
266C22×C4#9911yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C4⋊C411yescharacteristic
270C22×C4C2×C4⋊C411yescharacteristic
271M16—88no—
272C2×C8—44no—
273C2×C8—44no—
274C2×C8—84no—
275C2×C8—82no—
276C2×C8—82no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—416no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—44no—
293C42—48no—
294C42—44no—
295C42—84no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—41no—
311C23—41no—
312C23—42no—
313C23—42no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—44no—
319C23—42no—
320C23—44no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—44no—
325C23—44no—
326C23—41no—
327C23—41no—
328C23—42no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—81no—
335C23—81no—
336C23—82no—
337C23—82no—
338C23C2×#9511yescharacteristic
339C23C4×M1612yes—
340C23#40712yes—
341C23C2×#9911yescharacteristic
342C23C2×#9911yescharacteristic
343C23#41312yes—
344C23#42311yescharacteristic
345C23#42311yescharacteristic
346D4—48no—
347D4—44no—
348D4—44no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—28no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—48no—
371C2×C4—44no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4#34212yes—
403C2×C4#34212yes—
404C2×C4AΣL(1,𝔽16)12yes—
405C2×C4#35212yes—
406C8—84no—
407C8—84no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—42no—
420C22—41no—
421C22—41no—
422C22—44no—
423C22—44no—
424C22—42no—
425C22—42no—
426C22—44no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—82no—
439C22—84no—
440C22—82no—
441C22—82no—
442C22#146512yes—
443C22#146512yes—
444C22#172811yescharacteristic
445C22C2×#34211yescharacteristic
446C22#174211yescharacteristic
447C22C2×AΣL(1,𝔽16)11yescharacteristic
448C22C2×#35211yescharacteristic
449C22#212012yes—
450C4—22no—
451C4—48no—
452C4—41no—
453C4—42no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—42no—
458C4—84no—
459C4#139014yes—
460C2—22no—
461C2—21no—
462C2—21no—
463C2—42no—
464C2—41no—
465C2—41no—
466C2—82no—
467C2#791111yescharacteristic
468C2#1152512yes—
469C2#1152612yes—
470C2C2×#139011yescharacteristic
471C2#1257311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7187C2semidirect
13. #1466C4semidirect
15. #1465C4semidirect
104. C23×C4M16semidirect
136. C2×C22⋊C4M16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#14662
C2×#14652
C4×#3862
#71872